Manufacturing processes for producing products usually rely on quantitative measurements to provide information required for process control. Such measurements can be made on the final product, and/or on intermediate stages of the product within the manufacturing process, and/or on tools/fixtures used in the manufacturing process. For example, in semiconductor chip fabrication, measurements can be performed on finished chips (i.e., final product), on a wafer patterned with a photoresist (i.e., intermediate stage), or on a mask (i.e., a tool or fixture). Frequently, as in the case of semiconductor chip fabrication, these measurements are performed on structures having small dimensions. Furthermore, it is highly desirable to perform process control measurements quickly and non-destructively, in order to ensure a minimal impact on the process being controlled. Since optical measurements can be performed quickly, tend to be non-destructive, and can be sensitive to small features, various optical process control measurements have been developed.
Optical process control measurements can often be regarded as methods for measuring parameters of a pattern. For example, a pattern can be a periodic one-dimensional or two-dimensional grating on the surface of a wafer, and the parameters to measure can include feature dimensions, feature spacings and depth of the grating. To measure these parameters, an optical response of the pattern is measured. For example, reflectance as a function of wavelength can be measured. Typically, the optical response will depend on the parameter (or parameters) of interest in a complicated way such that direct parameter extraction from measured data is impractical. Instead, a mathematical model is typically constructed for the pattern, having the parameters of interest as variables. Within the model, a modeled optical response is calculated corresponding to the measured optical response. The parameters of interest are then determined by adjusting the variables to fit the modeled response to the measured response. Various optical process control measurements differ depending on the measured response(s), and on the kind of mathematical model employed.
A commonly-employed modeling approach for grating diffraction, known as the rigorous coupled wave analysis (RCWA), is described by Moharam et al. in Journal of the Optical Society of America (JOSA), A12, n5, p 1068-1076, 1995. The RCWA was first introduced by K. Knop in JOSA, v68, p 1206, 1978, and was later greatly improved by Moharam et al. in the above-referenced article. Some implementations of the RCWA for 1-D gratings are described in U.S. Pat. No. 6,590,656, U.S. 6,483,580, U.S. 5,963,329, and U.S. 5,867,276. The RCWA has been extended to 2-D gratings e.g., as considered by Han et al., Applied Optics 31(13), pp 2343-2352, 1992; and by Lalanne in JOSA A 14(7), pp 1592-1598, 1997. The use of RCWA modeling for characterizing 2-D gratings is also considered in U.S. 2004/007173.
Since a grating is periodic, grating-diffracted optical fields can be expressed as a superposition of space harmonics, each space harmonic having a different spatial period. The RCWA proceeds by including a finite number of space harmonics in the analysis (e.g., M for a 1-D grating and MN for a 2-D grating). Increasing M or MN increases accuracy, but requires more computation time, while decreasing M or MN decreases computation time, but provides reduced accuracy. The space harmonics each correspond to a diffraction order, so in a typical 2-D case where positive diffraction orders 1 through Nx, negative diffraction orders −1 through −Nx, and zero order diffraction are to be included for the x-direction in a calculation, we have M=2Nx+1. Similarly N=2Ny+1 if Ny positive and negative orders are included for the y direction.
The time required to perform numerical RCWA calculations is dominated by matrix operations having a calculation time on the order of M3 for a 1-D grating or (MN)3 for a 2-D grating. Accordingly, various special cases have been considered in the literature where calculation time can be reduced compared to a more general case without reducing accuracy.
For example, in the above-referenced article by Moharam et al., 1-D planar diffraction is identified as a special case of 1-D conical diffraction. In 1-D planar diffraction, the plane of incidence of the light on the grating is perpendicular to the grating lines, while in 1-D conical diffraction, the plane of incidence makes an arbitrary angle with respect to the grating lines. Moharam et al. show that a 1-D planar diffraction calculation for N orders requires less than half the computation time of a 1-D conical diffraction calculation for N orders. Moharam et al. also indicate that for 1-D planar diffraction from a symmetric 1-D grating, the matrices to be processed take on special forms (i.e., symmetric for lossless gratings and Hermitian for lossy gratings), which can reduce computation time. Thus, a 1-D planar diffraction geometry has typically been used for grating characterization based on RCWA calculations.
Another special case for characterization with RCWA calculations which has been considered is normal incident angle illumination of a symmetric 1-D grating, e.g., as considered in U.S. Pat. No. 6,898,537. This case is especially simple, since illumination with normal incident angle is a special case of planar diffraction (i.e., the polarization coupling of conical diffraction does not occur), and illumination with normal incident angle on a symmetric grating leads to symmetric positive and negative diffraction orders. Thus N positive orders, N negative orders and the zero order can be accounted for in this case with only M=N+1 space harmonics. To accomplish this, a specialized RCWA assuming normal incident angle with a symmetric grating is derived from the standard RCWA.
However, the approach of U.S. Pat. No. 6,898,537 requires illumination with normal incident angle on the grating, which leads to practical difficulties. For example, in the common case where the response of interest is a zero order reflection, normal incident angle illumination requires separation of the incident light from the zero order reflected light. Providing such separation (e.g., with a beam splitter) requires additional optical element(s), which undesirably increases system complexity.
A symmetry-reduced RCWA for a 1-D grating illuminated at off-normal incidence such that the plane of incidence is parallel to the grating lines is described by the present inventors in the above-referenced application Ser. No. 10/940,243. In this method, symmetry is exploited to account for N positive and N negative diffraction orders (and zero order diffraction) with N+1 space harmonics.
Since 2-D RCWA calculations tend to be more time consuming than 1-D RCWA calculations, methods of reducing calculation time are of special interest for the 2-D case. For example, the above-referenced U.S. 2004/0078173 application considers the use of a library for storing intermediate results for improving efficiency. However, exploiting symmetry to reduce 2-D RCWA calculation time does not appear to be considered in the prior art. Thus it would be an advance in the art to provide characterization of 2-D gratings with a symmetry-reduced RCWA having decreased calculation time compared to a conventional 2-D RCWA.